Bob Pare' made the excellent point that not only size but quality is relevant. I definitely agree with the spirit of his remarks. Bob happens to have used in passing the term 'syntactic'. For clarity, the use of that term needs to be sharpened to avoid misunderstanding. Actually, the term 'syntax' refers NOT to small categories such as algebraic theories or rings, but rather to their PRESENTATION by signatures or by polynomial generators, et cetera. The process of presentation is an adjoint pair quite distinct from the semantical adjoint pair: both adjoint pairs have a category of theories or of rings in common but are otherwise quite independent. In particular, syntax is NOT the adjoint of semantics. Cratylus, Chomsky, and their 21st century followers can be refuted by looking soberly at the actual practice of mathematics (wherein the construction of sequences of words and of diagrams is pursued with great care for the purpose of communication. That syntax is only remotely dependent on the structure of the content that is to be communicated). Both of the functors ?--------------> theories -------------->Large categories Syntax Semantics are needed. The domain category of the first can be chosen in various useful ways: sketches or diagrams of signatures et cetera. Happy new year! Bill On Fri 01/01/10 9:48 AM , pare@mathstat.dal.ca (Robert Pare) sent:
I would like to add a few thoughts to the "evil" discussion.
My 30+ years involvement with indexed categories have led me to the following understanding. There are two kinds of categories, small and large (surprise!). But the difference is not mainly one of size. Rather it's how well we can pin down the objects. The distinction between sets and classes is often thought of in terms of size but Russell's problem with the set of all sets was not one of size but rather of the nature of sets. Once you think you have the set of all sets, you can construct another set which you had missed. I.e. the notion is changing, slippery. There are set theories where you can have a subclass of a set which is not a set (c.f. Vopenka, e.g.)Smallness is more a question of representability: a functor may fail to be representable because it's too big (no solution set) or, more often, because it's badly behaved (doesn't preserve products, say). Subfunctorsof representables are not usually representable.
In our work on indexed categories, Schumacher and I had tried to treat this question by considering categories equipped with a groupoid of isomorphisms, which we called *canonical*, and then consider functors defined up to canonical isomorphism. In small categories only identitieswere canonical whereas in large categories, all isomorphisms were canonical.Our ideas were a bit naive and not well developed and earned us some ridicule,so we quietly stopped talking about it. Recently, Makkai developed an extensive theory of functors defined up to isomorphisms, FOLDS, but did not consider the possibility of specifying which isomorphisms ahead of time, so small categories were not included.
When I used to teach category theory, before Dalhousie made me chuck my chalk chuck, I would tell students there were two kinds of categories inpractice. Large ones which are categories of structures, corresponding tovarious branches of mathematics we wished to study. These categories supported various universal constructions, all defined up to isomorphism.Two large categories are considered to be the same if they are equivalent.It was considered impolite to ask if two objects were equal. Then there are the small categories which are used to study the large ones. These are syntactic in nature. For these, one can't expect the kinds of universal constructions that large categories have, but now it's okay, even necessary, to consider equality between objects. I went on to say that there were then four kinds of functors. Functors between large categorieswere to be thought of as constructions of one structure from another, e.g.the group ring. Functors between small categories were interpretations ofone theory in another or reindexing or rearranging. Functors from small to large categories were models or diagrams in the large one. These kindsof functors are perhaps the most important of the four, although this maybe debatable. The fourth kind, from large to small are rarer. They can be thought of as gradings or partitions of the large category.
Well, after these ramblings, perhaps my message is lost. So here it is: Small categories -> equality of objects okay Large categories -> equality of objects not okay Small is beautiful, not evil.
Bob
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